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Tannaka duality makes two complementary statements: first, that an affine algebraic group can be recovered from its category of representations (the "reconstruction problem"), and second, that certain categories are the categories of representations of an affine algebraic group (the "recognition problem").
Another kind of answer is Atiyah duality: if we write $D(X)$ for the Spanier-Whitehead dual of $X$ then $D(M/\partial M)$ is the Thom spectrum $M^{-TM}$. For any finite spectrum $X$ we have $H_k(DX)=H^{-k}(X)$ and $H^k(DX)=H_{-k}(X)$. For any finite complex $X$ and any orientable virtual vector bundle $V$ of virtual dimension $d$ we also have $H^k(X^V)=H^{k-d}(X)$ and $H_k(X^V)=H_{k-d}(X)$. If $M$ is an orientable manifold of dimension $n$ this gives $H^{k-n}(M^{-TM})=H^k(M)$. By combining these facts we can recover Poincar duality from Atiyah duality.
_______________ The assertion concerning Grothendieck's conjectureC13 “We shall see in the main body of the chapter that it is aconsequence of various duality theorems...” is misleading since (atbest) this is true for finite residue fields. According to a lecture ofSiegfried Bosch (20.10.04), the status of the conjecture over a discretevaluation ring R is as follows. When the residue field k is perfect, itis known when R has mixed characteristic (0,p)(Bgeuri), k is finite (McCallum), A is potentially totallydegenerate (i.e., after an extension of the field its reduction is atorus) (Bosch), or A is a Jacobian (Bosch and Lorenzini); it is stillopen when K is of equicharacteristic p>0 and the residue field isinfinite. For k nonperfect, the conjecture fails. The first exampleswere found by Bertapelle and Bosch, and Bosch and Lorenzini found manyexamples among Jacobians.
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We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself. In this sequence, the singular homology H_* maps to linearized contact cohomology CH^* which maps to linearized contact homology CH_* which maps to singular homology. In particular, the sequence implies a duality between the kernel of the map (CH_*\to H_*) and the cokernel of the map (H_* \to CH^*). Furthermore, this duality is compatible with Poincare duality in L in the following sense: the Poincare dual of a singular class which is the image of a in CH_* maps to a class \alpha in CH^* such that \alpha(a)=1. The exact sequence generalizes the duality for Legendrian knots in Euclidean 3-space [24] and leads to a refinement of the Arnold Conjecture for double points of an exact Lagrangian admitting a Legendrian lift with linearizable contact homology, first proved in [6].
In this paper we continue the study (initiated in a previous article) of linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras. We construct this duality in a very general setting, and prove its compatibility with morphisms of vector bundles and base change.
We study dimensional trends in ground states for soft-matter systems. Specifically, using a high-dimensional version of Parrinello-Rahman dynamics, we investigate the behavior of the Gaussian core model in up to eight dimensions. The results include unexpected geometric structures, with surprising anisotropy as well as formal duality relations. These duality relations suggest that the Gaussian core model possesses unexplored symmetries, and they have implications for a broad range of soft-core potentials. 5376163bf9